Generating the symmetric group by three prefix reversals

Abstract

The cubic pancake graphs are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate $\mathrm{Sym}_n$. We present a partial answer to this problem, in particular, we characterize all generating sets of three elements that contain at least one of the prefix reversals $r_2, r_3, r_{n-2}$, and $r_{n-1}$. We also give some computational results relating to the diameter and the girth of some cubic pancake graphs.

Figures (3)

• Figure 1: Visualization of Conjecture \ref{['conj:approx']}. The figure depicts the number $f(n)$ of pairs $(m,k)$ so that $\langle r_n,r_m,r_k\rangle=\mathop{\mathrm{\mathrm{Sym}_n}}\nolimits$ for several values of $n$. Four colors are used representing the residue $r$ of $n$ modulo 4. The curves represent the best rational model $(x^2 + u x + v)/q$: $r=0$ (black): $q=13$, $u=-0.5$, $v=5$, $\mathrm{RMSE}=3.343$; $r=1$ (green): $q=9$, $u=5$, $v=-45$, $\mathrm{RMSE}=12.132$; $r=2$ (blue): $q=10$, $u=-1$, $v=0$, $\mathrm{RMSE}=11.408$; $r=3$ (red): $q=7$, $u=3$, $v=-14$, $\mathrm{RMSE}=38.677$.
• Figure 2: Visualization of Conjecture \ref{['conj:mod4']} to illustrate that as $n$ grows, the previous triples are "preserved" $\pmod{4}$. The left-most figure depicts all pairs $(m,k)$ so that $\langle r_n,r_m,r_k\rangle=\mathop{\mathrm{\mathrm{Sym}_n}}\nolimits$. These pairs give rise to generating triples for $n=25+4\times 5$ by adding $4\times5$ to each index, and these are depicted in the middle figure (shown in black). There are other generating triples that are not inherited from the $n=45$ case (shown in pink). The right-most figure is built similarly by adding $4\times 8$ to each index for the triples for $n=45$. We see again that the previous pairs in black and pink are preserved for $n=77$ by adding $4\times8$ to each index.
• Figure 3: Visualization of Conjecture \ref{['conj:mod2']} to show on how the $n$ case is "embedded" to the $n+2$ case. In the first pictures in the left-most column, we depict all generating triples for $n=96$. All of these triples give rise to generating triples for $n=98$ by adding $2$ to each index (depicted in pink). The figures in the right-most column are built in a similar fashion.

Theorems & Definitions (27)

• Lemma 1: W64
• Lemma 2
• Lemma 3: Janusz
• Lemma 4: IradmusaTaleb
• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 5
• proof